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JetCalculus[DifferentialEquationData] - create a data structure for a system of differential equations

Calling Sequences

     DifferentialEquationData(EqList, VarList)

Parameters

     EqList    - a list of functions on a jet space defining a system of differential equations

     VarList   - a list of jet space coordinates such that the equations in EqList may be solved algebraically for these coordinates

 

Description

Examples

Description

• 

In the jet bundle approach to the geometric theory of differential equations, a system of k-th order differential equations is represented by a sub-manifold Rk of a jet space JkE, where π :E M is a fiber bundle. A solution to the differential equation defined by the sub-manifold Rk is a section s:M  E whose jets jksx take values in Rk.

• 

The DifferentialEquationData command creates an internal data structure which allows for the subsequent manipulation of the system of differential equations. The differential equations can be formally prolonged to higher order jet spaces using the Prolong command. The imbedding φ:RkJkEcan be constructed using the Transformation command.

• 

The command DifferentialEquationData is part of the DifferentialGeometry:-JetCalculus package.  It can be used in the form DifferentialEquationData(...) only after executing the commands with(DifferentialGeometry) and with(JetCalculus), but can always be used by executing DifferentialGeometry:-JetCalculus:-DifferentialEquationData(...).

Examples

withDifferentialGeometry:withJetCalculus:

 

Example 1.

Create a space of 2 independent variables and 1 dependent variable.

DGsetupx,y,u,E,1:

 

Define the differential equation data structure for the Sine-Gordon equation uxy  = sinu.

E > 

DEDifferentialEquationDatau1,2=sinu[],u1,2

DEu1,2,u1,2sinu

(2.1)

 

Let's prolong the equation once. We get 3 equations which can be solved for the derivatives uxy =u1,1,  uyyx  = u2, 1, uxyy =u1,2.

E > 

DE1ProlongDE,1

DE1u1,2,u1,1,2,u1,2,2,u1,2sinu,cosuu1+u1,1,2,cosuu2+u1,2,2

(2.2)

 

Now we can turn DE1 into a transformation.

E > 

ιTransformationDE1

ιx=x,y=y,u=u,u1=u1,u2=u2,u1,1=u1,1,u1,2=sinu,u2,2=u2,2,u1,1,1=u1,1,1,u1,1,2=cosuu1,u1,2,2=cosuu2,u2,2,2=u2,2,2

(2.3)

 

Show that the function f vanishes on solutions to the Sine-Gordon equation.

E > 

fcosu[]u1u2+u1,22u1,2sinu[]+u2u1,1,2

fcosuu1u2+u1,22u1,2sinu+u2u1,1,2

(2.4)
E > 

Pullbackι,f

0

(2.5)

See Also

DifferentialGeometry

JetCalculus

Prolong

Pullback

Transformation

PDEtools

 


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